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a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 | import OAI.Probability.InvariantIsing.Pressure.RandomThermalSqueeze
namespace OAI
/-! Continuous square-root transport with uniform integrability. -/
noncomputable section
open MeasureTheory Filter
open scoped Topology ENNReal
namespace InvariantIsing
lemma uniformIntegrable_add_real {Ω : Type*} [MeasurableSpace Ω]
(P : Measure Ω) (X Y : ℕ → Ω → ℝ)
(hX : UniformIntegrable X 1 P) (hY : UniformIntegrable Y 1 P) :
UniformIntegrable (fun n ω => X n ω+Y n ω) 1 P := by
refine ⟨hX.1.add hY.1 le_rfl,?_⟩
obtain ⟨C,hC⟩ := hX.2
obtain ⟨D,hD⟩ := hY.2
refine ⟨C+D,fun n => ?_⟩
exact (eLpNorm_add_le (f := X n) (g := Y n) le_rfl).trans (add_le_add (hC n) (hD n))
lemma uniformIntegrable_abs_real {Ω : Type*} [MeasurableSpace Ω]
(P : Measure Ω) (X : ℕ → Ω → ℝ) (hX : UniformIntegrable X 1 P) :
UniformIntegrable (fun n ω => |X n ω|) 1 P := by
apply uniformIntegrable_of_abs_le P _ X
(fun n => (hX.aestronglyMeasurable n).norm) hX
intro n
filter_upwards [] with ω
simp only [Real.norm_eq_abs,abs_abs,le_refl]
lemma random_sqrt_scaled_limit {Ω : Type*} [MeasurableSpace Ω]
(P : Measure Ω) [IsProbabilityMeasure P] (X : ℕ → Ω → ℝ) (L a : ℝ)
(ha : |a| ≤ 2) (hX : ∀ n, Measurable (X n)) (hUI : UniformIntegrable X 1 P)
(hprob : TendstoInMeasure P X atTop (fun _ => L)) :
TendstoInMeasure P (fun n ω => Real.sqrt (a*X n ω)) atTop (fun _ => Real.sqrt (a*L)) ∧
Tendsto (fun n => eLpNorm (fun ω => Real.sqrt (a*X n ω)-Real.sqrt (a*L)) 1 P)
atTop (𝓝 0) := by
have hs : UniformIntegrable (fun n ω => 1+|X n ω|) 1 P :=
uniformIntegrable_add_real P _ _ (uniformIntegrable_const le_rfl (by simp) (memLp_const 1))
(uniformIntegrable_abs_real P X hUI)
have hsq : UniformIntegrable (fun n ω => Real.sqrt (a*X n ω)) 1 P := by
apply uniformIntegrable_of_abs_le P _ _
(fun n => (((hX n).const_mul a).sqrt).aestronglyMeasurable) hs
intro n
filter_upwards [] with ω
rw [abs_of_nonneg (Real.sqrt_nonneg _),abs_of_nonneg (by positivity : 0 ≤ 1+|X n ω|)]
apply (Real.sqrt_le_iff).mpr
constructor
· positivity
· have hax : a*X n ω ≤ 2*|X n ω| := (le_abs_self (a*X n ω)).trans (by
rw [abs_mul]
exact mul_le_mul_of_nonneg_right ha (abs_nonneg _))
nlinarith [sq_nonneg (|X n ω|),sq_abs (X n ω)]
have hp := tendstoInMeasure_continuous_comp P X L hX hprob
(fun x => Real.sqrt (a*x)) (Real.continuous_sqrt.comp (continuous_const.mul continuous_id))
exact ⟨hp,(pressure_L1_of_probability_and_uniformIntegrability P _ _ hsq hp).1⟩
end InvariantIsing
end
end OAI
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